StruggleIsReal

Let X be an exponential random variable with parameter λ. Find P(|X − μ| ≥ 3σ), where μ = E[X] and σ = SD(X).

Let X be an normal random variable with parameters (μ,σ^2).

Find P (|X − μ| ≤ σ).

Suppose that 20 points are selected at random and independently from the interval (−3, 2). This means that each point is selected from a uniform distribution with parameters (−3, 2).

A) What's the expected value of the first point selected?

B) What's the variance of the first point selected?

C) How many of the points can be expected to be less than 1?

The lifetime of a tire selected randomly from a used tire shop is 10,000•X miles, where X is a random variable with the probability density function

f(x)= {6/(x^2) if 2 < x < 3, 0 otherwise

A) What's the probability that a randomly selected tire at this shop lasts fewer than 25,000 miles?

B) Suppose that a randomly selected tire at this shop lasts fewer than 25,000 miles. What's the probability that this tire lasts between 22,500 and 25,000 miles?

Suppose that X is a random variable with the probability density function

f(x) = {e^-x if x ≥ 0, 0 otherwise

What is the probability density function of the random variable Y = ln(X)?

A box contains 700 blue and 300 red balls. 200 balls are randomly chosen at the same time, & X is the number of red balls chosen.

A) List the possible values that the random variable X can have.

B) Find the probability mass function of X.

C) Find E[X].

D) Find Var(X).

E) What's the probability that exactly 50 of the balls chosen are red?

Suppose that X is a discrete random variable with E[X] = 2 and E[X(X − 4)] = 3.

Compute Var(−5X + 6) and SD(−5X + 6).

Suppose that X is a discrete random variable with E[X] = 2 and E[X(X − 4)] = 3.

Compute Var(−5X + 6) and SD(−5X + 6).

A message sent over a particular communication channel has probability p of being distorted. The message is sent repeatedly until it is transmitted clearly, and each attempt takes 1 minute to process.

What's the probability that the transmission takes at least t minutes? Assume that t is a positive integer. (Your answer should be in terms of p and t.)

Suppose a random variable X can only take on the values −1, 0, 1, and 3, with probabilities being, respectively, 1/10, 1/5, C, and 3/10.

(a) Find C.

(b) Draw a graph of the probability mass function.

(c) Find the cumulative distribution function of X.

(d) Draw a graph of the cumulative distribution function of X.

Die 1 has 4 red & 2 blue faces. Die 2 has 2 red & 4 blue faces. A fair coin is flipped once. If it lands on heads, the game continues with Die 1; if it lands on tails, then Die 2 is used.

(a) If the first two throws result in red, what is the probability of red on the third throw?

(b) If red turns up on the first two throws, what is the probability that Die 1 is being used?

4 balls are to be randomly selected, without replacement, from an urn that contains 15 balls numbered 1 through 15.

Let X be the number on the largest numbered ball selected. What are the possible values of X and the probabilities associated with them?

Let p(x) = x/10, x = 1, 2, 3, 4 be a probability mass function of a random variable X.

Determine F, the cumulative distribution function of X, and sketch a graph of the cumulative distribution function of X.

Each of 2 balls is painted either red or blue and then placed in an urn. suppose that each ball is colored blue with probability 1/2 & that these events are independent.

A) suppose that you know that red paint has been used (and so at least 1 of the balls is painted red). Compute the conditional probability that both balls are red.

Let A, B, and C be independent events. Prove that

P(A ∪ B ∪ C) = 1 − (1 − P(A))(1 − P(B))(1 − P(C)) either using De Morgan’s laws or expanding the right-hand side product and using the principle of inclusion-exclusion.

An urn contains 1 red and 2 blue balls. Balls are drawn from the urn successively, at random and with replacement. What's the probability that exactly 3 blues occur in the first 5 trials?

Suppose you have a wallet that contains either a $1 bill or a $10 bill (with equal probability). You add a $1 bill. Later you reach into your wallet and randomly remove a bill. It's a $1 bill. What's the probability that the bill remaining in the wallet is a $1 bill?

One of the cards of an ordinary deck of 52 cards is lost. (Each card has an equal probability of being the lost card.) What's the probability that a random card drawn from this deck is an ace?